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One-sided type-D Ricci-flat multicenter metrics

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371

Phys. Rev. D 114, 044041 – Published 13 August, 2026

DOI: https://doi.org/10.1103/bwp1-9wcp

Abstract

We employ a simplified form of Tod’s ansatz—which is built on the LeBrun-Tod ansatz—to construct one-sided type-D Ricci-flat multicenter metrics. These metrics are Hermitian non-Kähler and conformally Kähler, and they are determined by a pair of generating potentials: an axisymmetric harmonic function (different from the one originally proposed by Tod) in an auxiliary 3D flat space and its “harmonic conjugate.” We carry out a systematic study of these multicenter metrics, including their rod structure, asymptotic structure, and recovering from them various known closed-form examples. In particular, we show that, when fitted into the scheme of multisoliton solutions on flat space constructed by the present author using the inverse-scattering method, these metrics, with number of centers $n \leq 3$ , have a free-soliton number 2 or 3 greater than their phantom-soliton number—we conjecture that this is true for general $n$ .

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General relativity equations & solutions

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References (39)

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One-sided type-D Ricci-flat multicenter metrics

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